Consider the Sch ·· 0dinger equation
i ∂Ψ/∂t = − ∂2Ψ/∂x2, Ψ = Ψ(x, t),
which holds for all values of x and all positive values of t. Let the initial value of Ψ(x, t) be
Ψ(x, 0) = f(x)
and let the boundary conditions for Ψ be
Ψ(±∞, t) = 0.
(a) Make a Fourier transform of the Sch ·· 0dinger equation with respect to x and solve the resulting ordinary differential equation.
(b) Express the solution Ψ(x, t) satisfying the initial condition (B) in the form of a Fourier integral.